Question

In Exercises \(17-22,\) specify (a) the period, (b) the amplitude, and (c) identify the viewing window that is shown. $$y=5 \sin \frac{x}{2}$$

Short answer

The amplitude of the given function \(y=5 \sin \frac{x}{2}\) is 5. The period of the function is \(4\pi\). A suggested viewing window is \([-4\pi, 4\pi, -5, 5]\)

Step-by-step solution

Determine the Amplitude

The amplitude of a sine function is the absolute value of the coefficient of the sine function. In this case, the amplitude of the function is \(|5|\), which simplifies to 5.

Determine the Period

The period of a sine function is calculated as \(2\pi\) divided by the absolute value of the constant multiplied with the function's variable, in this case \(x\). Here the function's variable \(x\) is multiplied by \(1/2\), so the period of the function \(y = 5\sin \frac{x}{2}\) is \(2\pi\) divided by \(|1/2|\), which simplifies to \(4\pi\).

Identifying the Viewing Window

The exact specification of the viewing window depends on the typical range and domain for a sine function. A sine function has no fixed range for the x-values, hence normally \(x\) is in \(-\infty,+\infty\). As for the y-values, they are controlled by the amplitude. As the amplitude is 5, the maximum and minimum value of \(y\) should be 5 and -5 respectively. Therefore, a proper viewing window could be \([-4\pi, 4\pi, -5, 5]\). In this interval, we can see at least one full period of the sine function.